Homomorphisms of Hecke-Kiselman Monoids Associated to Simple Oriented Graphs

Abstract

Let $\Gamma$ and $\Theta$ be finite simple oriented graphs, and let $\mathbf{HK}_{\Gamma}$ and $\mathbf{HK}_{\Theta}$ denote the corresponding Hecke-Kiselman monoids. We present a set of Boolean matrices that is in bijection with the set of monoid homomorphisms from $\mathbf{HK}_{\Gamma}$ to $\mathbf{HK}_{\Theta}$. In particular, we obtain a Boolean matrix monoid isomorphic to the endomorphism monoid of $\mathbf{HK}_{\Theta}$.